In open systems, mass can fl ow into and out of the system. A generic open system with two streams in and two streams out is shown in Figure 2.9. In setting up the balance equations, it is convenient to discuss rates: mass fl ow rates [kg/s] and rates of energy transfer [J/s or W]. We must keep track of the mass in the system since it can change with time. In the general case of many streams in and many streams out, we must sum over all the in and out streams. We fi rst consider conservation of mass, for a nonreacting system.
The accumulation of mass in the system is equal to the difference of the total rate of mass in minus the total rate of mass out. Thus the mass balance can be written:
adm
dtb
sys5 ain m#
in2 aoutm#
out (2.16)
where m# is the mass fl ow rate in [kg/s]. For a stream fl owing through a cross-sectional area A, at velocity VS, the mass fl ow rate can be written as:
m# 5 AVS
v^ (2.17)
In many situations, it is convenient to express the amount of chemical species in terms of moles (molar basis) rather than mass. Mass can be converted to moles using the molecular weight.
The energy balance for an open system contains all the terms associated with an energy balance for a closed system, but we must also account for the energy change in the system associated with the streams fl owing into and out of the system. To accomplish this task, we consider the case of the generic open system illustrated in Figure 2.9. This open system happens to have two streams in and two streams out; however, the balances developed here will be true for any number of inlet or outlet streams.
Let’s look at the contribution to the energy balance from the inlet stream labeled stream 1. Compared to the closed-system analysis we performed in Section 2.4, there are Flow Work
Figure 2.9 Schematic of an open system with two streams in and two streams out. The piston shown in the plot is hypothetical; it illustrates the point that flow work is always associated with fluid flowing into or out of the system.
In in
System out in Stream 2
Stream 1
out Imaginary
piston δ(Eflow)in=− PinAinδx PE= Pin
min mout
Qsys
Ws
ûin
(êK)in (êP)in
ˆwflow= (PVˆ)in ˆ
ûout
(êK)out (êP)out
ˆwflow= (PV)out
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2.5 The First Law of Thermodynamics for Open Systems ◄ 61 two additional ways in which energy can be transferred from the surroundings. First, the molecules fl owing into the system carry their own energy; typically the most important form of energy is the specifi c internal energy, u^in, but the fl owing streams can also have (macroscopic) kinetic energy, 1e^k2in, and potential energy, 1e^P2in, associated with them.
Inspection of Equations (2.2) and (2.3) shows these last two terms can be written as 12 VS2 and gz respectively. Second, the inlet stream adds energy into the system by supplying work, the so-called flow work. Flow work is the work the inlet fl uid must do on the sys- tem to displace fl uid within the system so that it can enter. It may be visualized by placing an imaginary piston in front of the material that is about to enter the system, as depicted in Figure 2.9. The imaginary piston acts like the real piston shown in Figure 2.5. The rate of work exerted by the fl uid to enter the system is, therefore, given by:
1W#
flow2in5 2PE Ain
dx
dt 5Pin AinVSin5Pin m#
inv^in
where Equation (2.17) was used. We have set PE5Pin and eliminated the negative sign, since the direction of velocity is the negative of the normal vector from the piston. If we divide by mass fl ow rate, we can write the fl ow work in intensive form:
1w^flow2in5 1W#
flow2in
m#
in 5Pinv^in
We conclude that the fl ow work of any inlet stream is given by the term 1Pv^2in.
By a similar argument, we can show the fl ow work of any outlet stream is given by:
1W#
flow2out5 2m#
outPoutv^out
We can write the total energy transfer due to work in the system in terms of shaft work, Ws, and fl ow work, as follows:
W# 5W#
shaft1W#
flow5 BW#
s1 ain m#
in1Pv^2in1 aoutm#
out12Pv^2outR
The shaft work is representative of the useful work that is obtained from the system.
While shafts are commonly used to get work out of an open system, as discussed earlier, W#
s generically represents any possible way to achieve useful work; therefore, it does not include fl ow work! The fl ow work does not provide a source of power; it is merely the
“cost” of pushing fl uid into or out of the open system.
We can now include the new ways in which energy can exchange between the sys- tem and surroundings in our energy balance for open systems. In the general case of many streams in and many streams out, we must sum over all the in and out streams.
First, we consider a system at steady-state; that is, there is no accumulation of energy or mass in the system with time. The energy balance is written as [on a rate basis, by analogy to Equation (2.15)]:
05 ain m#
in(u^ 1 1
2VS21gz)in2 aoutm#out(u^1 1
2VS21gz)out energy fl owing
into the system with the inlet steams steady state
energy fl owing out of the system with the outlet steams
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62 ► Chapter 2. The First Law of Thermodynamics
1Q# 1 BW#
s1 ain m#
in1Pv^2in1 aoutm#
out12Pv^2outR
Rearranging, we get:
05 ain m#
inC1u^ 1Pv^2 11
2VS21gzDin2 aoutm#
outC1u^ 1Pv^2 1 1
2VS21gzDout1Q# 1W#
s
Enthalpy
The inlet streams that fl ow into open systems always have terms associated with both internal energy and fl ow work; therefore, it is convenient to group these terms together (so that we don’t forget one). We give it the name enthalpy, h [J/mol]:
h;u1Pv (2.18)
Since u, P, and v are all properties, this new group, enthalpy, is also a property. Thus, the additional energy associated with the fl owing inlet stream is given by h^in [as well as 1
2VS2 and gz]. The term h^in includes both the internal energy of the stream and the fl ow work it adds to enter the system. Similarly, the combined internal energy and fl ow work leaving the system as a result of the exiting streams are given by h^out.
Enthalpy provides us a property that is a convenient way to account for these two contributions of fl owing streams to the energy in open systems. As we will learn in the next section, enthalpy also is convenient to use with closed systems at constant pressure.
We next develop the relationship between temperature and enthalpy for an ideal gas. Recall the internal energy of an ideal gas depends only on T. Application of the defi - nition of enthalpy and the defi nition of an ideal gas gives:
hideal gas5u1Pv5u1RT5f1T only2
since Pv5RT for an ideal gas. Thus, like the internal energy, the enthalpy of an ideal gas depends only on T.
Steady-State Energy Balances
In summary, the steady-state energy balances can be written:
05 ain m#
in(h^ 1 1
2VS21gz)in2 aoutm#
out(h^ 11
2VS21gz)out
1Q# 1W#
s (2.19)
Take a look at Equation (2.19). See if you can identify the physical meaning of each term.
In cases where we can neglect (macroscopic) kinetic and potential energy, the steady-state, integral energy balance becomes:
05 ain m#
inh^in2 aoutm#
outh^out1Q# 1W#
s (2.20)
fl ow work from the inlet streams
fl ow work from the outlet streams
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Another useful form of the energy balance for open systems is for unsteady-state con- ditions. Unsteady-state is important, for example, in start-up as the equipment “warms up.” In the case when the inlet and outlet streams stay constant with time, the unsteady- state energy balance becomes:
¢dU dt 1 dEK
dt 1 dEP
dt ≤
sys5 ain m#
inah^ 1 1
2VS21gzb
in2 aoutm#
outah^ 11
2VS21gzb
out
1Q# 1W#
s (2.21)
In cases where we can neglect (macroscopic) kinetic and potential energy, the unsteady-state, integral energy balance becomes:
adU dtb
sys5 ain m#
inh^in2 aoutm#
outh^out1Q# 1W#
s (2.22)
In Equations (2.21) and (2.22), the left-hand side represents accumulation of energy within the system. There is no fl ow work associated with this term; hence, the appropri- ate property is U. On the other hand, the fi rst two terms on the right-hand side account for energy fl owing in and out of the system, respectively. These terms must account for both the internal energy and fl ow work of the fl owing streams. In this case, h^ is appropri- ate. It is worthwhile for you to take a moment and reconcile the use of U and h^ above. It will save you many mistakes down the road!